Two resistors connected in series have an equivalent resistance of 783.4 Ω. When they are connected in parallel, their equivalent resistance is 171.3 Ω. Find the resistance of each resistor.

Ω (small resistance)
Ω (large resistance

Equation 1.)

R1+R2=783.4

Equation 2.)

R1*R2/(R1+R2)=Req=783.4

Substitute equation 1 into 2:

R1=783.4-R2

and

R1*R2/(R1+R2)=Req=171.3

(783.4-R2)*R2/783.4-R2+R2=171.3

R2^2-783.4R2=-1.342 x 10^5

R2^2 -783.4R2+1.342 x 10^5=0

(R2 -3.663 x 10^2)^2

R2=3.663 x 10^2 Ω

R1 + R2=783.4 Ω

R1=783.4 Ω-R2

R1=734.4Ω-3.663 x 10^2 Ω

R1=417.1 Ω

To find the resistance of each resistor, we can use the formulas for the equivalent resistance of resistors connected in series and parallel.

Let's assume the resistance of the first resistor is R1 (small resistance) and the resistance of the second resistor is R2 (large resistance).

When two resistors are connected in series, the equivalent resistance (Rs) is the sum of the individual resistances:
Rs = R1 + R2

We are given that the equivalent resistance of the series connection is 783.4 Ω, so we have:
783.4 Ω = R1 + R2 ...........(Equation 1)

When two resistors are connected in parallel, the equivalent resistance (Rp) is given by the formula:
1/Rp = 1/R1 + 1/R2

We are given that the equivalent resistance of the parallel connection is 171.3 Ω, so we have:
1/171.3 = 1/R1 + 1/R2 ...........(Equation 2)

Now we have a system of equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of R1 and R2.

To eliminate the fractions in Equation 2, we can take the reciprocal of both sides:
171.3 = R1R2 / (R1 + R2) ...........(Equation 3)

Now we have three equations:
783.4 = R1 + R2 ...........(Equation 1)
171.3 = R1R2 / (R1 + R2) ...........(Equation 3)

We can solve these equations to find the values of R1 and R2. There are various ways to do this, such as substitution or graphing, but one common method is elimination.

Multiply Equation 1 by R1 and Equation 3 by (R1 + R2):
783.4R1 = R1(R1 + R2) ...........(Equation 4)
171.3(R1 + R2) = R1R2 ...........(Equation 5)

Expand Equation 4:
783.4R1 = R1^2 + R1R2 ...........(Equation 6)

Substitute Equation 5 into Equation 6:
783.4R1 = R1^2 + 171.3(R1 + R2) ...........(Equation 7)

Distribute 171.3 to get:
783.4R1 = R1^2 + 171.3R1 + 171.3R2 ...........(Equation 8)

Combine like terms:
R1^2 + 912.7R1 - 783.4R1 - 171.3R2 = 0
R1^2 + 129.3R1 - 171.3R2 = 0 ...........(Equation 9)

Now we have a quadratic equation in R1 and R2. We can solve this equation to find the values of R1 and R2 using the quadratic formula.

The quadratic formula is given by:
R = (-b ± √(b^2 - 4ac))/(2a)

In our case, the quadratic equation is:
R1^2 + 129.3R1 - 171.3R2 = 0

Comparing it with the general form of a quadratic equation: ax^2 + bx + c = 0, we get:
a = 1, b = 129.3, c = -171.3R2

Using the quadratic formula, we can find R1 in terms of R2.

Once we solve for R1, we can substitute its value back into Equation 1 to find R2.